# Unit 4 Contextual Applicationsap Calculus

Unit 1: Limits and Continuity Unit 2: Derivatives Unit 3: Existence Theorems Unit 4: Using Derivatives to Analyze Functions Unit 5: Applications of Derivatives Unit 6: Accumulation and Riemann Sums Unit 7: Antiderivatives and the Fundamental Theorem of Calculus Unit 8: Applications of Definite Integrals. Day 4: Rates of Change in Applied Contexts Other than Motion. Day 5: Review Day. Day 6: Quiz (Topics 4.1-4.3) Day 7: Intro to Related Rates. Day 8: More Advanced Related Rates. Day 9: Related Rates (Continued) Day 10: Quiz (Topics 4.4-4.5) Day 11: Approximating Values of a Function Using Linearization. Day 12: L’Hospital’s Rule. Therefore, dydt=12(−60)3√4=−1203√dydt=12(−60)34=−1203 miles per hour. 4.0 Unit 4 Overview: Contextual Applications of Differentiation Now that you have mastered the rules and formulas of differentiation, it is time for us to apply them! This section focuses on taking all the previous derivative rules and applying them in different contexts.

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ENDURING UNDERSTANDING

CHA-3 Derivatives allow us to solve real-world problems involving rates of change.

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 Topic Name Essential Knowledge 4.1 Interpreting the Meaning of the Derivative in ContextLEARNING OBJECTIVECHA-3.A Interpret the meaning of a derivative in context. CHA-3.A.1 The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable. CHA-3.A.2 The derivative can be used to express information about rates of change in applied contexts. CHA-3.A.3 The unit for is the unit for f divided by the unit for x.

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At Just the Right Time A good problem

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 4.2 Straight Line Motion: Connecting Position, Velocity, and AccelerationLEARNING OBJECTIVECHA-3.B Calculate rates of change in applied contexts. CHA-3.B.1 The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.

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The Ubiquitous Particle Motion Problem – a PowerPoint Presentation and its Handout

Motion Problems: Same Thing Different Context (11-16-2012) Matching Motion (9-16-2016)

Motion Matching A quick quiz

Speed (11-19-2012)

Speed Activity An exploration on Speed

A Note on Speed (4-21-2018) An analytic approach

Brian Leonard’s Particle Motion Game Velocity Game and answers Velocity game Answers

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 4.3 Rates of Change in Applied Contexts Other than MotionLEARNING OBJECTIVECHA-3.C Interpret rates of change in applied contexts. CHA-3.C.1 The derivative can be used to solve problems involving rates of change in applied contexts.

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 4.4 Introduction to Related RatesLEARNING OBJECTIVECHA-3.D Calculate related rates in applied contexts. CHA-3.D.1 The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable. CHA-3.D.2 Other differentiation rules, such as the product rule and the quotient rule, may also be necessary to differentiate all variables with respect to the same independent variable.

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### Unit 4 Contextual Applicationsap Calculus 2

Related Rates Problems 1

Related Rate Problems II

Good Question 9 Baseball and Related Rates

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 4.5 Solving Related Rate ProblemsLEARNING OBJECTIVECHA-3.E Interpret related rates in applied contexts. CHA-3.E.1 The derivative can be used to solve related rates problems; that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.

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Related Rates Problems 1

Related Rate Problems II

Good Question 9 Baseball and Related Rates

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 4.6 Approximating Values of a Function Using Local Linearity and LinearizationLEARNING OBJECTIVECHA-3.F Approximate a value on a curve using the equation of a tangent line. CHA-3.F.1 The tangent line is the graph of a locally linear approximation of the function near the point of tangency. CHA-3.F.2 For a tangent line approximation, the function’s behavior near the point of tangency may determine whether a tangent line value is an underestimate or an overestimate of the corresponding function value.

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### Unit 4 Contextual Applicationsap Calculus Pdf

Local Linearity The graphical manifestation of the derivative

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ENDURING UNDERSTANDING

LIM-4 L’Hospital’s Rule allows us to determine the limits of some indeterminate forms.

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 4.7 Using L’Hospital’s Rule for Determining Limits of Indeterminate FormsLEARNING OBJECTIVELIM-4.A Determine limits of functions that result in indeterminate forms. LIM-4.A.1 When the ratio of two functions tends to or in the limit, such forms are said to be indeterminate. LIM-4.A.2 Limits of the indeterminate forms or may be evaluated using L’Hospital’s Rule.

EXCLUSION STATEMENT: There are many other indeterminate forms, such as , for example, but these will not be assessed on either the AP Calculus AB or BC Exam. However, teachers may include these topics, if time permits.

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Determining the Indeterminate 1

Determining the Indeterminate 2 Same name, different post. Examining an implicit relation

Locally Linear L’Hôpital Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

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